a minimal solution for relative pose with unknown focal length
DESCRIPTION
A Minimal Solution for Relative Pose with Unknown Focal Length. Henrik Stewenius, David Nister, Fredrik Kahl, Frederik Schaffalitzky Presented by Zuzana Kukelova. Six-point solver (Stew énius et al ) – posing the problem. The linear equations from the epipolar constraint - PowerPoint PPT PresentationTRANSCRIPT
Center for Machine PerceptionDepartment of Cybernetics, Faculty of Electrical Engineering
Czech Technical University in Prague
A Minimal Solution for Relative Pose with Unknown Focal
Length
Henrik Stewenius, David Nister, Fredrik Kahl, Frederik Schaffalitzky
Presented by Zuzana Kukelova
Zuzana Kúkelová [email protected]
2/11
Six-point solver (Stewénius et al) – posing the problem
The linear equations from the epipolar constraint
Parameterize the fundamental matrix with three unknowns
Fi – basic vectors of the null-space
Solve for F up to scale => x = 1
0 1,..,6Ti im Fm i
1 2 3F xF yF zF
Zuzana Kúkelová [email protected]
3/11
Six-point solver (Stewénius et al) – posing the problem
Substitute this representation of F into the rank constraint
and the trace constraint
where and
2 0T TEE E trace EE E
det 0F
TK FK E 2
1 0 0
0 1 0 ,
0 0
Q w f
w
2 0T TFQF QF trace FQF Q F
Zuzana Kúkelová [email protected]
4/11
Six-point solver (Stewénius et al) – posing the problem
10 polynomial equations in 3 unknowns – y,z,w (1 cubic and 9 of degree 5)
10 equations can be written in a matrix form
where M is a 10x33 coefficient matrix and X is a vector of 33 monomials
. 0,M X
Zuzana Kúkelová [email protected]
5/11
Six-point solver (Stewénius et al) - computing the Gröbner basis
Compute the Gröbner basis using Gröbner basis elimination procedure Generate polynomials from the ideal
Add these polynomials to the set of original polynomial equations
Perform Gauss-Jordan elimination
Repeat and stop when a complete Gröbner basis is obtained
These computations (Gröbner basis elimination procedure) can be once made in a finite prime field to speed them up - offline
The same solver (the same sequence of eliminations) can be then applied to the original problem in - online
p
Zuzana Kúkelová [email protected]
6/11
Six-point solver (Stewénius et al)- elimination procedure
9 equations from trace constraint and , and .
detw F det F 2 detw F
Zuzana Kúkelová [email protected]
7/11
Six-point solver (Stewénius et al)- elimination procedure
The previous system after a Gauss-Jordan step and adding new equations based on multiples of the previous equations.
Zuzana Kúkelová [email protected]
8/11
Six-point solver (Stewénius et al)- elimination procedure
The previous system after a Gauss-Jordan step and adding new equations based on multiples of the previous equations.
Zuzana Kúkelová [email protected]
9/11
Six-point solver (Stewénius et al)- elimination procedure
Gauss-Jordan eliminated version of the previous system. This set of equations is a Gröbner basis.
Zuzana Kúkelová [email protected]
10/11
Six-point solver (Stewénius et al)- action matrix
Construction of the 15x15 action matrix for multiplication by one of the unknowns extracting the correct elements from the eliminated 18x33 matrix and
organizing them
zM
Zuzana Kúkelová [email protected]
11/11
Six-point solver (Stewénius et al)- extract solutions
The eigenvectors of the action matrix give solutions for
Using back-substitution we obtain solutions for F and f
We obtain 15 complex solutions
, ,y z w